什么是分形？
What is a fractal?
分形是具有自相似结构的几何图形，即局部放大后与整体形状相似，且通常具有非整数的维度。著名的曼德博集合和科赫雪花曲线都是分形例子。
A fractal is a geometric shape with self-similar structure, meaning a magnified part resembles the whole, and it often has a non-integer dimension. Famous examples include the Mandelbrot set and the Koch snowflake curve.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
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B. B. Mandelbrot《The Fractal Geometry of Nature》, Wiley, 1982
B. B. Mandelbrot, The Fractal Geometry of Nature, Wiley, 1982
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780716711865 (1982) · 核验依据 2 ISBN 9780716711865 (2021)
#分形 #自相似 #维度
为什么要引入‘四维时空’而不是只用三维空间？
Why introduce 'four-dimensional spacetime' instead of just 3D space?
狭义相对论把时间当作与空间平权的第四维，事件用(x,y,z,t)描述；洛伦兹变换本质是在这个四维空间里的‘旋转’。这统一了运动与时间的相对性。
Special relativity treats time as a fourth dimension on par with space, describing events by (x,y,z,t); Lorentz transformations are essentially 'rotations' in this 4D space, unifying motion and relativistic time.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
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A. 爱因斯坦 1905 年狭义相对论论文；S. 温伯格《引力和宇宙学》相关章节
A. Einstein (1905) special relativity paper; S. Weinberg, 'Gravitation and Cosmology', on time dilation
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL3398933W：Gravitation and cosmolo
#高维 #相对论
非欧几何是怎么被发现的？它颠覆了什么常识？
How was non-Euclidean geometry discovered, and what intuition did it overturn?
数学家们长期试图从欧几里得其余公理推导出「平行公设」，最终罗巴切夫斯基和波尔约在 19 世纪独立发现：若否定该公设也能得到自洽的几何（双曲几何）。黎曼随后建立椭圆几何，其中不存在平行线。这些几何后来成为爱因斯坦广义相对论的时空模型基础，证明「空间弯曲」在逻辑与物理上都是可能的。
Mathematicians long tried to derive Euclid's parallel postulate from his other axioms; Lobachevsky and Bolyai independently found in the 19th century that denying it yields a consistent geometry (hyperbolic). Riemann then built elliptic geometry, where no parallels exist. These geometries later became the basis of spacetime in Einstein's general relativity, showing that 'curved space' is both logically and physically possible.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#非欧几何 #平行公设 #双曲几何 #广义相对论
巴拿赫-塔斯基悖论是什么？
What is the Banach-Tarski paradox?
在三维空间中，可以（借助选择公理）把一个实心球分成有限块，重新拼成两个与原球同样大小的球。它揭示无限集合与体积的诡异性。
A solid ball can be split and reassembled into two identical copies of itself, exploiting the axiom of choice and infinite sets.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #悖论
双曲几何是什么？
What is hyperbolic geometry?
双曲几何是“过直线外一点可作无数条平行线”的非欧几何，常用庞加莱圆盘模型可视化——圆盘内越远的地方被压缩得越厉害，三角形内角和小于 180°。
Hyperbolic geometry is non-Euclidean with infinitely many parallels through a point; in the Poincaré disk model, triangles have angle sum < 180°.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #双曲几何
卡拉比-丘流形是什么？
What is a Calabi–Yau manifold?
卡拉比-丘流形是带有特殊“里奇平坦”度量的紧致复流形，在弦论中充当额外维度的空间形状。它们有丰富的拓扑结构，连接代数几何与物理。
Calabi–Yau manifolds are compact complex manifolds with a Ricci-flat metric; in string theory they describe the shape of extra dimensions and link algebraic geometry to physics.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
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Yau,《Calabi-Yau manifold》,Scholarpedia,2009 (DOI: 10.4249/scholarpedia.6524)
Yau, 'Calabi-Yau manifold', Scholarpedia, 2009, DOI: 10.4249/scholarpedia.6524
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.4249/scholarpedia.6524
#几何 #卡拉比丘
里奇流是什么？
What is Ricci flow?
里奇流是一种让流形度量随“时间”演化、像热扩散一样抹平曲率的几何流。佩雷尔曼借助它证明了庞加莱猜想，获千禧年大奖。
Ricci flow evolves a manifold's metric over time like heat diffusion, smoothing curvature. Perelman used it to prove the Poincaré conjecture, earning a Millennium Prize.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：此条未指向带版次的可引用著作（通识性内容、机构文件或未展开的英文书名），本库不提供独立书目出处；如需精确引用，需由编辑补全具体篇目与版次。
Clay Mathematics Institute 庞加莱猜想专题
Clay Mathematics Institute Poincaré conjecture
#几何 #几何流 #庞加莱猜想
非欧几何是在放弃哪条古典公设后建立起来的，它有哪些类型？
After abandoning which classical postulate is non-Euclidean geometry built, and what types are there?
非欧几何否定了欧几里得的平行公设，即过直线外一点有唯一平行线，从而发展出双曲与椭圆两种几何。罗巴切夫斯基与波尔约建立双曲几何，黎曼建立椭圆几何，二者在曲面与相对论中都有用。格林伯格的教材梳理了这些公理体系的发展。
Non-Euclidean geometry denies Euclid's parallel postulate that exactly one parallel passes through a point off a line, yielding hyperbolic and elliptic geometries. Lobachevsky and Bolyai built hyperbolic geometry and Riemann built elliptic geometry, both useful in surfaces and relativity. Greenberg's text traces the development of these axiomatic systems.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Stillwell《Sources of Hyperbolic Geometry》
Greenberg, Euclidean and Non-Euclidean Geometries; Stillwell, Sources of Hyperbolic Geometry
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL3908982W：Euclidean and non-Eucli
#非欧几何 #平行公设 #双曲几何
巴拿赫-塔斯基悖论为什么能「把一个球变成两个同样大的球」？
Why can the Banach–Tarski paradox 'turn one ball into two identical balls'?
巴拿赫-塔斯基悖论（1924）指出：在三维空间中，可以把一个实心球分割成有限块，经刚体旋转和平移后重新拼成两个与原球完全相同大小的球。这个结论并不违反物理，因为它依赖选择公理构造出不可测的「病态」碎片，现实中无法真正切出。它揭示了选择公理与「体积」直觉之间的深刻冲突。
The Banach–Tarski paradox (1924) says a solid ball in 3D can be cut into finitely many pieces that, after rotations and translations, reassemble into two balls identical to the original. This doesn't violate physics because the pieces are non-measurable 'pathological' sets built using the Axiom of Choice and cannot be physically cut. It reveals a deep conflict between the Axiom of Choice and our intuition of volume.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#巴拿赫-塔斯基 #选择公理 #不可测集 #悖论
射影几何的对偶原理是什么？
What is the principle of duality in projective geometry?
射影几何研究在投影变换下不变的性质，其对偶原理说：把命题中的“点”与“直线”互换，定理依然成立。它统一了许多看似不同的几何结论。
Projective geometry studies properties invariant under projection; its duality principle says swapping 'point' and 'line' in a theorem yields another true theorem. It unifies many geometric facts.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #射影 #对偶
欧拉线是什么？
What is the Euler line?
在绝大多数非等边三角形中，重心、垂心、外心和九点圆圆心共线，这条直线称为欧拉线。等边三角形这些点重合，故无此线。
In most non-equilateral triangles the centroid, orthocenter, circumcenter and nine-point center are collinear on the Euler line.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #三角形
在球面上，三角形的内角和为什么可以大于 180°？大到多少？
On a sphere, why can a triangle's angle sum exceed 180°, and by how much?
球面上的“直线”是大圆，三个大圆围出的区域就是球面三角形。取球面上互相垂直的三条大圆（比如赤道和两条经线），围出的三角形三个角都是 90°，内角和 270°，超过 180° 的部分叫球面过剩。过剩量与三角形面积成正比（单位球上，面积恰等于过剩角的弧度数），这是球面几何里最深刻的定理之一，被用来从三角测量反推地球大小。
Great circles play the role of straight lines on a sphere, and three of them enclose a spherical triangle. Take three mutually perpendicular great circles (say the equator and two meridians): the triangle they enclose has three right angles, a sum of 270°. The excess over 180° is the spherical excess, and on the unit sphere it is exactly proportional to the triangle's area — in fact equal to it in radians. This deep result lets surveyors measure the size of the Earth from angle measurements.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-03 · 2 条独立证据
来源 · Source
《什么是数学》(柯朗)（球面几何）
What Is Mathematics? (Courant), spherical geometry
独立核验依据 · Independent verification: 核验依据 2 DOUBAN ISBN 标识 9787309128109（按标识回读: 什么是数学（原作名 What Is Ma
#球面几何 #球面过剩 #大圆
双曲几何有什么特点？
What characterizes hyperbolic geometry?
在双曲几何中，过直线外一点可作无穷多条与该直线平行（不相交）的线，三角形内角和小于180°。庞加莱圆盘模型以优美方式可视化它。
In hyperbolic geometry, infinitely many lines through a point avoid intersecting a given line, and triangle angles sum below 180°. The Poincaré disk model visualizes it elegantly.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #非欧 #曲率
黎曼流形是什么？
What is a Riemannian manifold?
黎曼流形是一个在每个点都配备内积（度量）的光滑曲面或空间，从而能定义长度、角度和曲率。它是广义相对论描述时空的数学载体。
A Riemannian manifold is a smooth space with an inner product at each point, defining lengths, angles and curvature. It is the mathematical stage of general relativity.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #流形 #度量
分形几何用什么概念度量像海岸线这样不规则物体的维数？
What concept does fractal geometry use to measure the dimension of irregular objects like coastlines?
分形几何用豪斯多夫维数等分数维刻画自相似且处处不光滑的结构，其维数可以介于整数之间。曼德博在1982年的著作中以海岸线为例说明，测量长度随尺度变化正源于这一非整数维。法尔科纳的《分形几何》系统建立了相关数学。
Fractal geometry uses fractional dimensions such as the Hausdorff dimension to describe self-similar, nowhere-smooth structures whose dimension can lie between integers. Mandelbrot's 1982 book used coastlines to show that measured length changes with scale precisely because of this non-integer dimension. Falconer's Fractal Geometry builds the mathematics systematically.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Mandelbrot《The Fractal Geometry of Nature》（1982）；Falconer《Fractal Geometry》
Mandelbrot, The Fractal Geometry of Nature (1982); Falconer, Fractal Geometry
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780716711865 (1982)
#分形 #豪斯多夫维数 #自相似
射影几何里的帕斯卡定理（Pascal's theorem）说内接于圆锥曲线的六点有什么共线关系？
What collinearity does Pascal's theorem assert for six points on a conic in projective geometry?
帕斯卡定理指出：若六边形（可退化为六点）的顶点都在同一条圆锥曲线上，则三组对边交点共线，这条线称为帕斯卡线。它是射影几何对“度量”完全不敏感的代表性结论，对偶于布里安雄定理（外切六边形的三线共点）。
Pascal's theorem states that if the six vertices of a hexagon lie on a single conic, then the three intersections of opposite sides are collinear—the Pascal line. It is a hallmark of projective geometry because it makes no use of distance or angle, and it is dual to Brianchon's theorem for circumscribed hexagons.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Coxeter, H. S. M., 《射影几何》(第2版), Springer
Coxeter, H. S. M., 'Projective Geometry' (2nd ed.), Springer
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780387965321 (1987)
#帕斯卡定理 #圆锥曲线 #射影几何 #对偶
等周不等式是什么？
What is the isoperimetric inequality?
在所有周长相同的平面图形中，圆围成的面积最大；等价地，面积相同时圆的周长最短。公式为 4πA ≤ L²。
Among closed planar curves of fixed length, the circle encloses maximal area: 4πA ≤ L², with equality only for a circle.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #不等式
平行公设在欧几里得体系中处于什么特殊地位？
What special status does the parallel postulate hold in the Euclidean system?
平行公设在欧几里得《几何原本》中比其他公设更复杂、更难凭直觉接受，长期被视为应被证明的定理。它的独立性最终促成非欧几何的诞生。格林伯格与希思的著作都详述了围绕它的两千年争论。
In Euclid's Elements the parallel postulate is more complex and less intuitively obvious than the others, long suspected of being a provable theorem. Its independence ultimately gave birth to non-Euclidean geometry. Greenberg and Heath detail the two-millennia debate around it.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#平行公设 #欧几里得 #公理独立性
只用无刻度直尺和圆规，为什么无法把任意角三等分、把立方体体积加倍，或作出与已知圆面积相等的正方形？这三个古希腊难题的“不可解”最终是如何被严格证明的？
Using only an unmarked straightedge and compass, why is it impossible to trisect an arbitrary angle, double the volume of a cube, or construct a square equal in area to a given circle — and how were these ancient Greek problems rigorously proven unsolvable?
关键是把可作图的量翻译成域论语言：从单位长度出发，每次作图至多添加一个二次扩张，所以任何可作图数在有理数域上的次数必为2的幂。三等分60°角等价于求方程 x^3-3x-1=0 的根，其最小多项式次数为3而非2的幂，故一般角不可三等分；倍立方归结为 x^3-2=0，同理被排除；化圆为方则因π是超越数而不可能。Wantzel于1837年发表了前两个问题的完整证明。
The key is translating constructibility into field theory: starting from a unit length, each construction step adds at most one quadratic extension, so every constructible number has degree a power of 2 over the rationals. Trisecting a 60-degree angle reduces to solving x^3-3x-1=0, whose minimal polynomial has degree 3 rather than a power of 2, while doubling the cube leads to x^3-2=0 and is excluded the same way; squaring the circle fails because pi is transcendental. Pierre Wantzel published complete proofs of the first two impossibilities in 1837.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
皮埃尔·万采尔《判断几何问题能否用尺规求解的研究》，载《纯粹与应用数学杂志》（Journal de Mathématiques Pures et Appliquées）第2卷，1837年
Pierre Wantzel, 'Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas', Journal de Mathématiques Pures et Appliquées, vol. 2 (1837)
#尺规作图 #域论判据
蒲丰投针实验是怎样只用随机抛针就估算出圆周率 π 的？
How does Buffon's needle experiment estimate π using only randomly dropped needles?
把长为 l、间距为 d（l≤d）的平行线，随机抛下长度为 l 的针，针与某条线相交的概率恰为 2l/(πd)。因此只要大量投针、统计相交频率，就能用 π ≈ 2l/(d·频率) 反推出 π——这是最早的几何概率与蒙特卡洛方法实例之一。
Dropping a needle of length l onto parallel lines spaced distance d apart (with l ≤ d), the probability it crosses a line is exactly 2l/(πd). Counting crossings over many trials gives π ≈ 2l/(d·frequency)—one of the earliest instances of geometric probability and a Monte Carlo method.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
Buffon, G. L., 《道德算术随笔》(Essai d'arithmétique morale), 1777
Buffon, G. L., 'Essai d'arithmétique morale', 1777
#蒲丰投针 #几何概率 #圆周率 #蒙特卡洛
分形与分数维是什么？
What are fractals and fractal dimension?
分形是在任意尺度下都呈现自相似细节的形状，如科赫雪花。它的“维数”往往不是整数，可用豪斯多夫维度量，海岸线就是自然界的分形例子。
Fractals show self-similar detail at every scale, like the Koch snowflake. Their dimension is often fractional, measured by Hausdorff dimension; coastlines are natural fractals.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #分形 #自相似
四色定理说的是什么？它是怎样被证明的？
What does the Four Color Theorem state, and how was it proved?
四色定理断言：任何平面地图都可以用至多四种颜色给相邻国家着色而不重复。它由格斯里在 1852 年提出，最终由阿佩尔和哈肯在 1976 年借助计算机穷举近 2000 种不可避免构型完成证明，成为首个重大「计算机辅助证明」，也曾引发关于证明可接受性的争论。
The Four Color Theorem asserts that any planar map can be colored with at most four colors so that adjacent regions differ. Posed by Guthrie in 1852, it was finally proved by Appel and Haken in 1976 by computer-enumerating nearly 2000 unavoidable configurations—the first major computer-assisted proof, which sparked debate about acceptability of proofs.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#四色定理 #地图着色 #计算机证明 #平面图
电脑屏幕上那些五彩斑斓的曼德博集图形，是怎么“画”出来的？
How are those colorful Mandelbrot-set images on computer screens actually drawn?
对平面上每个点 c，从 z=0 开始反复计算 z←z²+c。如果迭代序列一直不逃到无穷远，c 就属于曼德博集；如果它很快飞走，就按“逃逸的速度”涂上不同颜色。集合内部是一张“心形”加一圈圆泡，边界极其复杂——无论放大多少倍，仍然有无穷细节。
For each point c in the plane, start at z=0 and repeatedly compute z←z²+c. If the iteration never escapes to infinity, c belongs to the Mandelbrot set; if it flies away, points are colored by how fast they escape. The set's body is a cardioid with bulbs, and its boundary is staggeringly intricate — zoom in as far as you like and detail never runs out.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
B. B. Mandelbrot《The Fractal Geometry of Nature》, Wiley, 1982
B. B. Mandelbrot, The Fractal Geometry of Nature, Wiley, 1982
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780716711865 (1982) · 核验依据 2 ISBN 9780716711865 (2021)
#曼德博集 #迭代 #分形边界
为什么圆周长是 2πr？
Why is circumference 2πr?
周长与直径之比恒为 π，故 C=π·2r=2πr。
The ratio of circumference to diameter is π, so C=2πr.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#圆周率 #直径 #周长
非欧几何是什么？
What is non-Euclidean geometry?
非欧几何放弃“平行公设”，发展出双曲与椭圆几何：三角形内角和分别小于或大于180°。它的诞生揭示公理系统的相对性，并启发了广义相对论。
Non-Euclidean geometry drops the parallel postulate, yielding hyperbolic and elliptic geometries where triangle angle sums differ from 180°. It revealed the relativity of axioms and inspired general relativity.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #公理 #曲率
椭圆是什么？
What is an ellipse?
椭圆是平面上到两个定点（焦点）距离之和为常数的点的轨迹。它是拉长的圆，行星轨道就是椭圆。
An ellipse is the locus of points whose sum of distances to two foci is constant; planetary orbits are ellipses.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #椭圆
高斯-博内定理是什么？
What is the Gauss–Bonnet theorem?
高斯-博内定理把曲面的总高斯曲率积分与它的欧拉示性数联系起来：∫_M K dA = 2π χ(M)。它表明“局部弯曲”的总和由“整体的拓扑”决定。
Gauss–Bonnet links total Gaussian curvature to topology: ∫_M K dA = 2π χ(M), so global bending is fixed by the Euler characteristic.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #高斯博内
圆反演（inversive geometry）变换把直线和圆变成什么，它有什么用处？
Under circle inversion (inversive geometry), what do lines and circles become, and what is it good for?
以某圆心、半径 r 作反演：平面上点 P 映到同一射线上满足 OP·OP' = r² 的点 P'。该变换把过圆心的直线保持为直线，把不过圆心的直线变成过圆心的圆，圆仍映成圆或直线，且保持“共形”（角度不变）。它被用来把看似复杂的圆系问题转化为直线问题，也用于复分析与双曲几何。
Inverting in a circle of radius r sends a point P to P' on the same ray with OP·OP' = r². Lines through the center stay lines, lines not through the center become circles through the center, and circles map to circles or lines; angles are preserved (conformal). It lets one turn messy circle configurations into straight-line problems and appears in complex analysis and hyperbolic geometry.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Coxeter, H. S. M., 《几何学导引》(第2版), Wiley
Coxeter, H. S. M., 'Introduction to Geometry' (2nd ed.), Wiley
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL15235428W：Introduction to geomet
#圆反演 #反演几何 #共形 #双曲几何
曼德博集合是什么？它的边界为什么如此神奇？
What is the Mandelbrot set, and why is its boundary so remarkable?
曼德博集合由复平面上使迭代 z → z² + c 保持有界的参数 c 构成，由本华·曼德博在 1980 年用计算机发现。它的边界具有无限精细的自相似细节——无论放大多少倍都会出现新的相似结构，是「分形」的典型代表。这种由极简单规则产生极端复杂图案的现象，深刻影响了混沌与复杂系统的研究。
The Mandelbrot set consists of complex parameters c for which the iteration z → z² + c remains bounded, discovered by Benoit Mandelbrot in 1980 using computers. Its boundary has infinitely detailed self-similar structure—zoom in any amount and new similar patterns appear—making it the icon of fractals. That a tiny rule generates extreme complexity reshaped the study of chaos and complex systems.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#曼德博集合 #分形 #自相似 #复迭代
勾股定理是什么？
What is the Pythagorean theorem?
勾股定理说直角三角形两直角边的平方和等于斜边平方：a² + b² = c²。例如边长3、4的直角三角形，斜边是5（9+16=25）。
The Pythagorean theorem states a² + b² = c² for the legs and hypotenuse of a right triangle; e.g. legs 3 and 4 give hypotenuse 5, since 9 + 16 = 25.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #定理
在一个平面上，过直线外一点真的只有一条平行线吗？如果不止一条呢？
Through a point off a line, is there really only one parallel? What if there were more than one?
欧几里得的第五公设说只有一条，但两千年来没人能证明它。19 世纪数学家发现：否定它并不会导致矛盾，而是得到全新的几何。在双曲几何里，过直线外一点可以有无数条平行线，三角形内角和小于 180°，空间像马鞍一样“每处都往里弯”（负曲率）。
Euclid's fifth postulate asserts exactly one parallel, yet for two millennia no one could prove it. In the 19th century mathematicians discovered that denying it creates no contradiction — only a new geometry. In hyperbolic geometry, infinitely many parallels pass through a point off a line, triangles have angle sums under 180°, and space curves inward like a saddle (negative curvature).
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-03 · 2 条独立证据
来源 · Source
《什么是数学》(柯朗)（非欧几何）
What Is Mathematics? (Courant), non-Euclidean geometry
独立核验依据 · Independent verification: 核验依据 2 DOUBAN ISBN 标识 9787309128109（按标识回读: 什么是数学（原作名 What Is Ma
#双曲几何 #平行公设 #非欧几何
为什么尺规作图不能三等分任意角？
Why can't an arbitrary angle be trisected with straightedge and compass?
尺规作图只能从已知数出发通过加减乘除和开平方构造新数，所以能作的点都在由有理数反复开平方扩出的域中。三等分 60° 角等价于解三次方程 4x³−3x=1/2（令 x=cos 20°），而这个方程没有“平方根式”的解，因此不可能。它与“化圆为方”“倍立方”并称古希腊三大作图难题，直到 19 世纪才借助域论被彻底解决。
Straightedge-and-compass constructions can build new numbers only via addition, subtraction, multiplication, division, and square roots, so constructible points lie in fields obtained from the rationals by repeated square roots. Trisecting 60° is equivalent to solving the cubic 4x³−3x=1/2 (with x=cos 20°), which has no solution built from square roots—hence it is impossible. Along with squaring the circle and doubling the cube, it is one of the three ancient Greek construction problems, fully settled only in the 19th century via field theory.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-03
来源 · Source
《什么是数学》(柯朗)（代数与几何章）
What Is Mathematics? (Courant), algebra and geometry chapter
#尺规作图 #三等分角 #域扩张
欧拉示性数（欧拉示性）是什么？
What is the Euler characteristic?
对多面体，欧拉示性数 χ = V − E + F（顶点减棱加面），对凸多面体恒为2。它推广到曲面后成为拓扑不变量，能区分不同形状的“洞数”。
For a polyhedron, χ = V − E + F (vertices minus edges plus faces) is always 2 for convex ones. Generalized to surfaces, it becomes a topological invariant counting holes.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #拓扑 #多面体
什么是圆锥曲线？
What are conic sections?
用平面去截一个双圆锥，得到的曲线包括圆、椭圆、抛物线和双曲线。它们在天体轨道和光学中都有重要应用。
Slicing a double cone with a plane yields circles, ellipses, parabolas and hyperbolas; they describe orbits and optics.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #圆锥曲线
为什么海岸线的长度会越量越长？它的维数为什么可以是分数？
Why does a coastline's measured length keep growing, and why can its dimension be fractional?
尺子取得越小，量出的海岸线就越长，因为在更精细的尺度下总会显现出新的弯曲细节，长度趋于发散而不收敛。曼德博提出用豪斯多夫维数刻画这种“粗糙度”：英国海岸线的维数约为 1.26，介于直线（1 维）与平面（2 维）之间，说明它“浓密”得介于两者之间。
The smaller the ruler, the longer the coastline, because new bends appear at every finer scale and the measured length keeps growing without converging. Mandelbrot proposed the Hausdorff dimension to quantify this roughness: Britain's coastline has dimension about 1.26, between a line (dimension 1) and a plane (dimension 2), capturing that it is denser than a curve yet thinner than a surface.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#豪斯多夫维数 #分形 #曼德博
为什么‘维度灾难’让高维数据变难？
Why does the 'curse of dimensionality' make high-dimensional data hard?
在高维中，单位体积里的大部分‘质量’都靠近边界，两点间距离趋于彼此接近，使聚类、采样和最近邻都失效或需要海量样本。
In high dimensions most of a unit volume's mass sits near the boundary and pairwise distances become nearly equal, breaking clustering, sampling and nearest-neighbor methods without huge data.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Goodfellow, I., Bengio, Y. & Courville, A.，《Deep Learning》，MIT Press，2016
Goodfellow, I., Bengio, Y. & Courville, A., Deep Learning, MIT Press, 2016
独立核验依据 · Independent verification: 核验依据 1 ISBN 978-0-262-03561-3 (MIT Press)
#高维 #距离
什么是重心（形心）？
What is the centroid of a shape?
重心是图形在均匀材质下平衡的支点，也是三角形三条中线的交点。均匀三角形的重心位于每条中线的三分之二处。
The centroid is the balance point of a uniform shape, the intersection of a triangle's medians, located 2/3 along each median.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #重心
什么是平面密铺（镶嵌）？
What is a tessellation (tiling)?
密铺是用同一种或几种图形无重叠、无缝隙地铺满平面。正三角形、正方形和正六边形可以单独密铺，正五边形则不行。
A tessellation tiles the plane with no gaps or overlaps; equilateral triangles, squares and regular hexagons each tile alone, pentagons do not.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #密铺
为什么三角形内角和是180度？
Why is triangle sum 180°?
过顶点作平行线，三内角恰等于平角180度。
A parallel line shows the three angles sum to a straight 180°.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#内角 #平行线 #平角
圆的面积公式是什么？
What is the area formula of a circle?
圆的面积是 A = π r²，其中 r 是半径。例如半径为2的圆面积约 12.57。
A circle's area is A=π r², where r is the radius; e.g. radius 2 gives about 12.57.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #圆
飞机从北京飞往纽约为什么绕北冰洋走弧线，而不是直飞？
Why does a flight from Beijing to New York curve over the Arctic instead of flying straight?
地球表面是球面，球面上两点之间的最短路径不是平面上的直线，而是经过这两点的大圆的弧——大圆是圆心与球心重合的大圆圈，比如赤道。北京到纽约的大圆弧刚好穿过北冰洋上空，所以航线图上看着像“绕远”，实际距离更短。
The Earth's surface is a sphere, and the shortest path between two points on it is not a straight line in the flat sense but the arc of a great circle — a circle centered at the sphere's center, like the equator. The great-circle arc from Beijing to New York passes over the Arctic, so on a flat map it looks like a detour but is actually the shorter route.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-03
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《什么是数学》(柯朗)（几何章：球面几何）
What Is Mathematics? (Courant), geometry (spherical geometry)
#球面几何 #大圆 #最短路径
欧拉多面体公式 V−E+F=2 是什么？
What is Euler's polyhedron formula V−E+F=2?
它表明任意凸多面体的顶点数V、棱数E与面数F满足 V−E+F=2。例如立方体：8−12+6=2。
Euler's formula V−E+F=2 holds for any convex polyhedron, e.g. a cube: 8−12+6=2.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #多面体
n 边形的内角和是多少？
What is the sum of interior angles of an n-gon?
n 边形内角和为 (n−2)×180°。例如四边形内角和为360°，五边形为540°。
The sum is (n−2)×180°. For a quadrilateral it's 360°, for a pentagon 540°.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #多边形
黄金比例约等于 1.618，它是怎么定义出来的？
The golden ratio is about 1.618 — how is it actually defined?
把一条线段分成长短两段，如果长段与短段的比等于整条线段与长段的比，这个比值就是黄金比例 φ。解方程 φ²=φ+1 得 φ=(1+√5)/2≈1.618。它同时是连续分数 1+1/(1+1/(1+…)) 的极限，也与正五边形对角线之比有关。
Split a segment into a longer and a shorter part; if the ratio of the longer to the shorter equals the ratio of the whole to the longer part, that ratio is the golden ratio φ. Solving φ²=φ+1 gives φ=(1+√5)/2≈1.618. It is also the limit of the continued fraction 1+1/(1+1/(1+…)) and appears in the diagonal ratio of a regular pentagon.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
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Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#黄金比例 #几何比例 #连续分数
什么是轴对称图形？
What is an axially symmetric figure?
如果一个图形沿一条直线对折后两边完全重合，它就是轴对称图形，这条直线叫对称轴，如正方形、圆、蝴蝶。
A shape is axially symmetric if it folds onto itself across a line (axis), like a square, circle or butterfly.
难度 进阶 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #对称
正多面体（柏拉图立体）一共有几种？
How many regular polyhedra (Platonic solids) are there?
只有5种：正四面体、正六面体（立方体）、正八面体、正十二面体和正二十面体。它们每个面都是全等的正多边形且每个顶点结构相同。
Exactly five: tetrahedron, cube, octahedron, dodecahedron, icosahedron—each face a congruent regular polygon with identical vertices.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #多面体
只许使用无刻度直尺与圆规，如何在一个给定射线的端点处作出一个角，使其等于已知的角∠AOB？写出作图步骤，并逐步说明所依据的原理。
Using only an unmarked straightedge and a pair of compasses, how do you construct, at the endpoint of a given ray, an angle equal to a given angle ∠AOB? Give the construction step by step and state the principle behind each step.
①以O为圆心、任意长r为半径作弧，交OA、OB于C、D；②在目标射线端点O′处以同一半径r作弧，弧上记一点C′；③以C′为圆心、CD之长为半径作弧，交前一弧于D′；④连接O′D′，则∠C′O′D′=∠AOB。原理在于△OCD与△O′C′D′三边对应相等（OC=O′C′=OD=O′D′=r，CD=C′D′），由SSS全等得对应角相等；这正是欧几里得《几何原本》第一卷命题23。
First draw an arc centered at O with an arbitrary radius r, cutting OA and OB at C and D. Second, with the same radius r draw an arc centered at the endpoint O′ of the target ray and mark a point C′ on it. Third, draw an arc centered at C′ with radius equal to CD, cutting the earlier arc at D′. Fourth, join O′D′; then ∠C′O′D′ = ∠AOB because triangles OCD and O′C′D′ have three pairs of equal sides (SSS congruence) — precisely Proposition I.23 of Euclid's Elements.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
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欧几里得《几何原本》（Euclid, Elements）第一卷命题23
Euclid, Elements, Book I, Proposition 23.
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#尺规作图 #角的复制
三角形内角和是多少度？
What is the sum of the interior angles of a triangle?
任意三角形的三个内角之和总是 180 度。例如等边三角形的每个角都是 60 度，加起来正好 180 度。
The three interior angles of any triangle always add up to 180 degrees. For example, each angle of an equilateral triangle is 60 degrees, and together they sum to 180 degrees.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
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Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#三角形 #内角和 #几何
球面三角学是什么？
What is spherical trigonometry?
球面三角学研究球面上由大圆弧构成的三角形。由于在球面上“直线”是大圆，其内角和大于 180°，常用球面余弦定理计算航海与天文距离。
Spherical trigonometry studies triangles on a sphere bounded by arcs of great circles; since straight lines on a sphere are represented by great circles, their angle sum exceeds 180 degrees, and the spherical law of cosines is commonly used to compute distances in navigation and astronomy.
难度 入门 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #球面三角
黄金比例 φ 是什么？它真的无处不在吗？
What is the golden ratio φ, and is it really everywhere?
黄金比例 φ ≈ 1.618 是方程 φ = 1 + 1/φ 的正解，即把一条线段分成大小两段，使整段与长段之比等于长段与短段之比。它确实出现在正五边形、斐波那契数列的极限比以及一些自然生长模式中，但许多关于它在艺术和人体中「普遍存在」的说法被过度渲染。它更多地是一个优美而真实的数论与几何常数。
The golden ratio φ ≈ 1.618 is the positive solution of φ = 1 + 1/φ, dividing a segment so the whole-to-large equals large-to-small. It genuinely appears in regular pentagons, the limiting ratio of Fibonacci numbers, and some natural growth patterns, but many claims of its ubiquity in art and the human body are exaggerated. It is best seen as an elegant, real constant of number theory and geometry.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#黄金比例 #斐波那契 #正五边形 #比例
什么是相似图形？
What is similarity of figures?
两个图形如果形状相同但大小可以不同，就称为相似，对应角相等、对应边成比例。例如地图上的图形与实地图形通常相似。
Two figures are called similar if they have the same shape but may differ in size, with equal corresponding angles and proportional corresponding sides. For example, a map and the real terrain it represents are usually similar.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#相似 #比例 #几何
什么是体积？
What is volume?
体积是三维物体所占空间的大小，常用立方单位（如立方米）度量。例如棱长为 a 的正方体体积是 a³。
Volume is the amount of space a three-dimensional object occupies, measured in cubic units such as cubic meters. For example, a cube with edge length a has volume a³.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#体积 #三维 #几何
什么是角度？
What is an angle?
角度是由两条有公共端点的射线所夹的部分，用来衡量两条线之间的张开程度，单位是度（°）或弧度。一个直角是 90 度，一个平角是 180 度。
An angle is the figure formed by two rays sharing a common endpoint, measuring how much the two lines open up; its unit is the degree or the radian. A right angle is 90 degrees and a straight angle is 180 degrees.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#角度 #度 #几何
圆周率π大约是多少？
What is the approximate value of pi?
圆周率π是圆周长与直径的比值，约等于3.14159，日常计算常用3.14来近似。
Pi is the ratio of a circle's circumference to its diameter, about 3.14159 (often 3.14).
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #圆周率
什么是平行四边形？
What is a parallelogram?
平行四边形是两组对边分别平行的四边形，它的对边相等、对角相等。常见的矩形和菱形都是特殊的平行四边形。
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs; its opposite sides are equal and its opposite angles are equal. Familiar rectangles and rhombuses are special kinds of parallelograms.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#平行四边形 #四边形 #几何
三角形有几条边？
How many sides does a triangle have?
三角形有3条边和3个角，是最简单的多边形。
A triangle has 3 sides and 3 angles; it's the simplest polygon.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《数学花园漫游记》,马希文,中国少年儿童出版社,2004
Ma Xiwen, Adventures in the Math Garden, China Children Press, 2004
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787500767558（豆瓣书目条目: 数学花园漫游记）
#几何 #图形
正方形有几条边？
How many sides does a square have?
正方形有4条一样长的边和4个直角。
A square has 4 equal sides and 4 right angles.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《数学花园漫游记》,马希文,中国少年儿童出版社,2004
Ma Xiwen, Adventures in the Math Garden, China Children Press, 2004
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787500767558（豆瓣书目条目: 数学花园漫游记）
#几何 #图形
什么是等距变换？
What is an isometry?
等距变换（刚性运动）保持任意两点间距离不变，如平移、旋转、反射。它不改变图形的形状和大小，只改变位置或朝向，是几何对称的基础概念。
An isometry (rigid motion) preserves all distances—translations, rotations, reflections. It keeps shape and size, only changing position/orientation.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #等距
柯西-施瓦茨不等式的几何意义是什么？
What is the geometric meaning of the Cauchy–Schwarz inequality?
在向量形式下，它说 |u·v| ≤ |u||v|，等号当且仅当两向量共线。几何上它表示两向量夹角的余弦绝对值不超过 1，是许多不等式与机器学习理论的根基。
In vector form |u·v| ≤ |u||v|, with equality iff the vectors are collinear; geometrically the cosine of their angle lies in [−1,1]. It underpins many inequalities.
难度 入门 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #柯西施瓦茨
莫比乌斯带是什么？
What is a Möbius strip?
莫比乌斯带是把纸条一端扭转180度再粘成环的曲面，它只有一个面、一条边。你沿着它一直画，会走完整条带子而不跨越边界。
A Möbius strip is made by twisting a paper strip 180° before joining the ends; it has only one side and one edge. Drawing along it covers the whole surface without crossing an edge.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #单侧曲面 #拓扑玩具
克莱因瓶是什么？
What is a Klein bottle?
克莱因瓶是一个没有内外之分、只有一个面的“瓶子”，在三维中必须自我穿透才能画出，但在四维空间里可以真正不相交地存在。它是单侧曲面的有趣例子。
A Klein bottle has no inside or outside and only one surface; in 3D it must intersect itself, but in 4D it exists without self-crossing. It is a famous one-sided surface.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #单侧曲面 #拓扑
什么是周长？
What is the perimeter?
周长是一个平面图形边界的总长度，也就是绕图形一圈的距离。例如正方形的周长是四条边相加，即 4 倍边长。
The perimeter is the total length of the boundary of a flat figure, that is, the distance around it. For example, the perimeter of a square is the sum of its four sides, or four times the side length.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#周长 #边界 #几何
什么是圆？
What is a circle?
圆是平面上到一个固定点（圆心）距离相等的所有点组成的集合，这个固定距离叫做半径。圆是最基本的曲线图形之一，常见于车轮、钟表等。
A circle is the set of all points in a plane that are at the same fixed distance from a given point (the center); that fixed distance is called the radius. The circle is one of the most basic curved shapes, seen in wheels and clocks.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#圆 #圆心 #半径
什么是长方形？
What is a rectangle?
长方形（矩形）有四条边、四个直角，对边长度相等。正方形是四边都相等的特殊长方形。
A rectangle has four right angles with opposite sides equal; a square is a special rectangle with all sides equal.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #图形
什么是正方形？
What is a square?
正方形是四条边相等且四个角都是直角的四边形，既是矩形也是菱形。正方形的面积等于边长乘边长，即 边长²。
A square is a quadrilateral with four equal sides and four right angles; it is both a rectangle and a rhombus. The area of a square equals its side length multiplied by itself, that is, side squared.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#正方形 #四边形 #面积
什么是角？
What is an angle?
角是由两条有公共端点的射线组成的图形，常用度数表示大小，直角是90°，平角是180°。
An angle is formed by two rays sharing an endpoint, measured in degrees; a right angle is 90°.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #角
圆是什么形状？
What shape is a circle?
圆是平面上到一个固定点（圆心）距离相等的所有点连成的闭合曲线，像车轮的边。
A circle is the set of points in a plane at a fixed distance from a center point, like the edge of a wheel.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#几何 #圆
为什么正多面体（柏拉图立体）只有五种？
Why are there only five regular (Platonic) solids?
正多面体要求每个面是全等的相同正多边形，且每个顶点汇聚相同数量的面。由于多边形内角必须使若干个能拼成小于 360° 的立体角，几何上只允许三角形、正方形和五边形这三种面，并仅能组合出正四面体、立方体、正八面体、正十二面体和正二十面体五种。柏拉图曾把它们与地水火风及宇宙第五元素相对应。
A regular solid requires identical regular polygons as faces meeting equally at each vertex. Because the polygon angles must sum to less than 360° to form a solid corner, only triangles, squares, and pentagons work, yielding exactly five: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Plato associated them with earth, water, fire, air, and the cosmos's fifth element.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
欧几里得《几何原本》第十三卷；大英百科全书（Britannica）
Euclid, Elements Book XIII; Encyclopaedia Britannica
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#柏拉图立体 #正多面体 #对称 #几何